Elementary mathematics
Based on "A Calculus of Number Based on Spatial Forms" by Jeffrey M. James 1993.
f1(x) = x
f2(x) = -x
f3(x) = ex
f4(x) = ln x
f3(f4(x)) = x eln x = x
f4(f3(x)) = x ln ex = x
<> - Inversion -0 = 0
() = o - Instanziation e0 = 1
[] = ∎ - Abstraction ln 0 = -∞ + [-∞...∞]i
[(a)] = ([a]) = a Involution Axiom 1 ln ea = eln a = a
(a[bc]) = (a[b])(a[c]) Distribution of a Axiom 2 ea + ln (b + c) = ea + ln b + ea + ln c
a<a> = Inversion of a Axiom 3 a - a = 0
<<a>> = a Inverse Cancellation (double Inversion) Theorem 1 -(-a) = a
<a><b> = <ab> Inverse Collection Theorem 2 (-a) + (-b) = -(a + b)
Proof of Inverse Cancellation for a != ∎
<<a>>
<<a>><a>a -Inversion of a
a Inversion of <a>
Proof of Inverse Collection for a,b != ∎
<a><b>
<a><b>ab<ab> -Inversion of ab
<ab> Inversion of a and b
Proof of additive self-inversion of void (ambiguous sign) -0 = 0
<>
Inversion of void
Cardinality of a (aggregation second degree)
a = ([a]) = ([a][o]) -Involution2, -Involution1
aa = ([a][o])([a][o]) = ([a][oo]) 2x a, -Distribution of [a]
a...a = ([a][o...o]) induction step to all natural numbers
a * b => ([a][b])
a / b => ([a]<[b]>) for b !=
Proof of Associativity of multiplication (a*b)*c = a*(b*c)
([([a] [b])][c] )
( [a] [b] [c] ) Involution1
( [a][([b] [c])]) -Involution1
Proof of multiplicative self-inversion of o 1/1 = 1
(<[o]>)
(< >) Involution1
( ) Inversion of void

Real part of the complex logarithm

Imaginary part of the complex logarithm - Riemann surface
Absorption
([a]∎) = Zero cardinality of a Theorem 3 a * 0 = 0
∎ a = ∎ Absorption of a (black hole) Theorem 4 ∎ + a = ∎
Proof of Zero cardinality
([a]∎)
([a]∎)([a][b])<([a][b])> -Inversion of ([a][b])
([a][b])<([a][b])> -Distribution of [a]
Inversion of ([a][b])
Proof of Absorption
∎ a
[(∎ [(a)])] 2x -Involution1
[ ] Zero cardinality of (a)
Proof of Zero cardinality
([a]∎)
( ∎) Absorption of [a]
Involution2
Indefiniteness of cardinality of ∎
∎ = ∎∎ = ∎∎∎ = ∎∎∎∎ = ... -Absorption of ∎
= <> = [o] = (∎) = [(<>)] = ([<>]) = <[o]> = <(∎)> 5x Inversion of void, 3x Involution1, 3x Involution2
Indefiniteness of Inversion of ∎
(<∎>) = ([o]<∎>) <= 1/0 is undefined (singularity)
0 * ? = 1 zero don't have multiplicative inverse
0/0 ambiguous
0/a = 0 a/a = 1 a/0 = ∞
inverse Absorption
<∎> a
<∎><<a>> -Inverse Cancellation
<∎ <a>> Inverse Collection
<∎ > Absorption of <a>
∎<∎> -Inversion of ∎
=> Absorption of <∎> or inverse Absorption of ∎
=> Possibility of arbitrary absorption of everything everywhere
=> <∎> undefined (Inversion of a black hole is not allowed)
=> Division by 0 is undefined (singularity)
Cardinality of [a] (aggregation third degree)
([a]...[a]) = (([[a]][o...o]))
Since multiplication is as commutative as addition, factors can be aggregated just like summands.
Addition 2 + 4 = 6
Multiplikation 2 * 4 = 2 + 2 + 2 + 2 = 8
Exponentiation 24 = 2 * 2 * 2 * 2 = 16
Tetration 42 = 2222 = 224 = 216 = 65.536
Hyper-5 24 = 2222 = 422 = 65.5362 = 2...2 65.536 times - therefore, our universe is far too small
...
Since exponentiation is not commutative, exponents cannot be represented aggregated like summands or factors.
Thus aggregations lose their beauty and thus their usefulness from here on.
oo ooo 2 + 3 = 5
( [oo] [ooo] ) = oo oo oo 2 * 3 = 2 + 2 + 2 = 6
(([[oo]] [ooo])) = ([oo][oo][oo]) 23 = 2 * 2 * 2 = 8
(([[oo]][ (([[oo]][oo])) ])) = (([[oo]][oooo])) 32 = 222 = 24 = 16
(([[oo]][ (([[oo]][ (([[oo]][oo])) ])) ])) 23 = 222 = 42 = 2222 = 65.536
...
... = 22 = 22 = 22 = 2*2 = 2+2 = 4
ab => (([[a]][b]))
a0 = 1 => (([[a]]∎)) = ((∎)) = o Absorption of [[a]], Involution2
a1 = a => (([[a]][o])) = (([[a]])) = ([a]) = a Involution1, 2x Involution2
Inversion
([<a>]b) = <([a]b)> Inverse Promotion1 Theorem 5 -a * eb = -(a * eb)
(<[<a>]>) = <(<[a]>)> Inverse Promotion2 Axiom 4 1/-a = -1/a
(<[<a>]>b) = <(<[a]>b)> Inverse Promotion3 Theorem 6 1/-a * eb = -(1/a * eb)
Proof of Inverse Promotion1 for a != ∎
([<a>]b)
<([a]b)>([a]b)([<a>]b) -Inversion of ([a]b)
<([a]b)>([a <a>]b) -Distribution von b
<([a]b)>([ ]b) Inversion of a
<([a]b)>([ ] ) Absorption von b
<([a]b)> Involution2
Proof of Inverse Promotion3 for a != ∎
( <[<a>]> b)
<(<[a]>b)>( <[a]> b)( <[<a>]> b) -Inversion of (<[a]>b)
<(<[a]>b)>([(<[a]>)]b)([ (<[<a>]>) ]b) 2x -Involution1
<(<[a]>b)>([(<[a]>) (<[<a>]>) ]b) -Distribution von b
<(<[a]>b)>([(<[a]>) <(<[ a ]>)>]b) Inverse Promotion2
<(<[a]>b)>([ ]b) Inversion of (<[a]>)
<(<[a]>b)>([ ] ) Absorption von b
<(<[a]>b)> Involution2
Proof of irrationality of √2
√2 = p/q and p, q ∈ ℤ shall be divisor extraneous
2 = (p/q)2
2q2 = p2
-> p2 is even
-> p is even
sei p = 2r für r ∈ ℤ
-> 2q2 = p2 = 4r2
-> q2 = 2r2
-> q is even
-> p und q are not divisor extraneous
=> √2 is irrational
(([[oo]]<[oo]>)) = ([p]<[q]>)
([[oo]]<[oo]>) = [p]<[q]>
q = o -> p = (([[oo]]<[ oo ]>))
p = o -> q = (([[oo]]<[<oo>]>))
q = oo -> p = (([[oo]]<[ oo ]>)[oo])
p = oo -> q = (([[oo]]<[<oo>]>)[oo])
q = ooo -> p = (([[oo]]<[ oo ]>)[ooo])
p = ooo -> q = (([[oo]]<[<oo>]>)[ooo])
...
-> p and q are not naturally resolvable together
=> (([[oo]]<[oo]>)) is irrational
Operators
-a => <a> for a != ∎
a+b => ab
a+b+c => abc
a-b => a<b> for b != ∎
a*b => ([a][b])
a*b*c => ([a][b][c])
a*-b => ([a][<b>]) = <([a][b])> = ([<a>][b]) +-Inverse Promotion1 for a,b != ∎
a/b => ([a]<[b]>) for b !=
1/b => ([o]<[b]>) = (<[b]>) Involution1 for b !=
a/a => ([a]<[a]>) = o Inversion of [a] for a !=
ab => (([[a]][b]))
a1 => (([[a]][o])) = (([[a]])) = a Involution1, 2x Involution2
a-b => (([[a]][<b>])) for b != ∎
a1/b => (([[a]]<[b]>)) = (([[a]][(<[b]>)])) -Involution1 for b !=
ac/b => (([[a]]<[b]>[c])) = (([[a]][(<[b]>[c])])) -Involution1 for b !=
abc
(([[a]][(([[b]][c]))]))
(([[a]] ([[b]][c]) )) Involution1
((ab)c)1/d = ab*c/d
(([[(([[(([[a]][b]))]][c]))]]<[d]>))
(( [[a]][b] [c] <[d]>)) 4x Involution1
ab * ac = ab+c
([(([[a]][b]))][(([[a]][c]))])
( ([[a]][b]) ([[a]][c]) ) 2x Involution1
( ([[a]][b c]) ) Distribution of [[a]]
ac * bc = (a*b)c
([(([[a]][c]))][(([[b]][c]))])
( ([[a]][c]) ([[b]][c]) ) 2x Involution1
( ([[a] [b]][c]) ) Distribution of [c]
1 / e = e-1 => ([o]<[(o)]>) = (<[(o)]>) = (<o>) 2x Involution1
1 / e2 = e-2 => ([o]<[(oo)]>) = (<[(oo)]>) = (<oo>) 2x Involution1
ln (a*b) = ln a + ln b => [([a][b])] = [a][b] Involution1
ln (a/b) = ln a - ln b => [([a]<[b]>)] = [a]<[b]> Involution1
ln (ab) = ln a * b => [(([[a]][b]))] = ([[a]][b]) Involution1
logb a = ln a/ln b => ([[a]]<[[b]]>)
1/(1/a) = a
(<[(<[a]>)]>)
(< <[a]> >) Involution1
( [a] ) Inverse Cancellation
a Involution2
(1/a)*(1/b) = 1/(a*b)
([(< [a]>)][(<[b] >)])
( < [a]> <[b] > ) 2x Involution1
( < [a] [b] > ) Inverse Collection
( <[([a] [b])]> ) -Involution1
(1/a)b = a-b
( ([[(<[a]>)]][ b ]) )
( ([ <[a]> ][ b ]) ) Involution1
(<([ [a] ][ b ])>) Inverse Promotion1
( ([ [a] ][<b>]) ) -Inverse Promotion1
a/c + b/d = (a*d + b*c) / c*d
( [a] <[c]>)( [b] <[d]>)
( [a][d] <[d]><[c]>)( [b][c] <[c]><[d]>) -Inversion of [d] and [c]
( [a][d] <[d] [c]>)( [b][c] <[c] [d]>) 2x Inverse Collection
([([a][d])]<[d] [c]>)([([b][c])]<[c] [d]>) 2x -Involution1
([([a][d]) ([b][c])]<[c] [d]>) -Distribution of <[c][d]>
1/a + 1/b = (a + b) / a*b
( <[a]>)( <[b]>)
([b]<[b]><[a]>)([a]<[a]><[b]>) -Inversion of [b] and [a]
([b]<[b] [a]>)([a]<[a] [b]>) 2x Inverse Collection
([ab]<[a][b]>) -Distribution of <[a][b]>
(a + b) * (a - b) = a² - b²
([a b][a <b>])
([a b][a]) ([a b][<b>]) Distribution of [ab]
([a][a])([b][a]) ([a][<b>]) ([b][<b>]) Distribution of [a] and [<b>]
([a][a])([b][a])<([a][ b ])><([b][ b ])> 2x -Inverse Promotion1
([a][a]) <([b][ b ])> Inversion of ([a][b])
(([[a]][oo])) <(([[b]][oo]))> Cardinality of [a] and [b]
Numbers
0 =>
1 => o
2 => oo
-1 => <o>
-2 => <oo>
1/2 => (<[oo]>)
2/3 => ([oo]<[ooo]>)
43 => ([b][oooo])ooo for b = oooooooooo
243 => ([b][([b][oo])oooo])ooo for b = oooooooooo
1243 => ([b][([b][boo])oooo])ooo for b = oooooooooo
Arity
{a} = ([oooooooooo][a]) Definition of {}
oooooooooo{a} = {o a} Carry+ Cardinality of oooooooooo
{a}{b} = {ab} Collection+
([{a}]b) = {([a]b)} Promotion+
{} = Zero cardinality+
23 * 114 = 2622
([ {oo}ooo ] [ {{o}o}oooo ])
([{oo}][{{o}o}oooo]) ([ooo][{{o}o}oooo]) Distribution of [{{o}o}oooo]
{([ oo ][{{o}o}oooo])} ([ooo][{{o}o}oooo]) Promotion+
{{{o}o}oooo{{o}o}oooo} {{o}o}oooo{{o}o}oooo{{o}o}oooo 2x -Cardinality of {{o}o}oooo
{{{o}o}oooo{{o}o}oooo} {{o}o}{{o}o}{{o}o}oooooooooooo Collection+ of o
{{{o}o}oooo{{o}o}oooo} {{o}o}{{o}o}{{o}oo} oo Carry+
{{{o}o} {{o}o} {o}{o}{o} oooooooooooo} oo Collection+ of {o}
{{{o}o} {{o}o} {o}{o}{oo} oo} oo Carry+
{{{o} {o} oooooo} oo} oo Collection+ of {{o}}
{{{ oo} oooooo} oo} oo Collection+ of {{{o}}}
Inverse Arity
/a\ = (<[oooooooooo]>[a]) Definition of /\
/oooooooooo a\ = o/a\ Carry- Cardinality of oooooooooo
/a\/b\ = /ab\ Collection-
([/a\]b) = /([a]b)\ Promotion-
/\ = Zero cardinality-
{/a\} = /{a}\ = a Arity Cancellation
12,1 * 1,012 = 12,2452
([ {o}oo/o\ ] [ o//o/oo\\\ ])
([{o}][o//o/oo\\\]) ([oo][o//o/oo\\\]) ([/o\][o//o/oo\\\]) 2x Distribution of [o//o/oo\\\]
{([ o ][o//o/oo\\\])}([oo][o//o/oo\\\])/([ o ][o//o/oo\\\])\ 2x Promotion ±
{ o//o/oo\\\ } o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ 3x -Cardinality of o//o/oo\\\
{o}{ //o/oo\\\ } o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ -Collection+
{o} /o/oo\\ o//o/oo\\\o//o/oo\\\ /o//o/oo\\\ \ Arity Cancellation1
{o}oo /o/oo\\ //o/oo\\\ //o/oo\\\ /o//o/oo\\\ \ Collection+ of o
{o}oo /oo/oo\ /o/oo\\ /o/oo\\ //o/oo\\\ \ Collection- of /o\
{o}oo /oo/oooo /oo\ /oo\ /o/oo\\\ \ Collection- of //o\\
{o}oo /oo/oooo /ooooo /oo\\\ \ Collection- of ///o\\\
100 / 3 = 33,3...
([ {{o}} ]<[ooo]>)
{([ {o} ]<[ooo]>)} Promotion+
{([oooooooooo]<[ooo]>)} -Carry+
{([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>) ([o]<[ooo]>)} 3x Distribution of <[ooo]>
{( )( )( ) ([o]<[ooo]>)} 3x Inversion of [ooo]
{ooo} {([o]<[ooo]>)} -Collection+
{ooo} ([{o}]<[ooo]>) -Promotion+
{ooo} ([oooooooooo]<[ooo]>) -Carry+
{ooo}([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>) ([o]<[ooo]>) 3x Distribution of <[ooo]>
{ooo}( )( )( ) ([o]<[ooo]>) 3x Inversion of [ooo]
{ooo}ooo ([/oooooooooo\]<[ooo]>) -Carry-
{ooo}ooo /([ oooooooooo ]<[ooo]>) \ Promotion-
{ooo}ooo/([ooo]<[ooo]>)([ooo]<[ooo]>)([ooo]<[ooo]>)([o]<[ooo]>) \ 3x Distribution of <[ooo]>
{ooo}ooo/( )( )( )([o]<[ooo]>) \ 3x Inversion of [ooo]
{ooo}ooo/ooo ([/oooooooooo\]<[ooo]>) \ -Carry-
{ooo}ooo/ooo /([ oooooooooo ]<[ooo]>)\\ Promotion-
...
100 / 3,3 = 30,3030...
([ {{o}} ]<[ooo/ooo\]>)
{([ {o} ]<[ooo/ooo\]>)} Promotion+
{([oooooooooo]<[ooo/ooo\]>)} -Carry+
{([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>)} -Carry- and 3x -Collection-
{([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito) ([ /o\ ]<[ooo/ooo\]>)} 3x Distribution of <[ooo/ooo\]>
{( )( )( ) ([ /o\ ]<[ooo/ooo\]>)} 3x Inversion of [ooo/ooo\]
{ooo} {([ /o\ ]<[ooo/ooo\]>)} -Collection+
{ooo} ([{/o\}]<[ooo/ooo\]>) -Promotion+
{ooo} ([ o ]<[ooo/ooo\]>) Arity Cancellation1
{ooo} ([/oooooooooo\]<[ooo/ooo\]>) -Carry-
{ooo} /([ oooooooooo ]<[ooo/ooo\]>) \ Promotion-
{ooo} /([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>) \ -Carry- and 3x -Collection-
{ooo}/([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito) ([/o\]<[ooo/ooo\]>) \ 3x Distribution of <[ooo/ooo\]>
{ooo}/( )( )( ) ([/o\]<[ooo/ooo\]>) \ 3x Inversion of [ooo/ooo\]
{ooo}/ooo ([//oooooooooo\\]<[ooo/ooo\]>) \ -Carry-
{ooo}/ooo //([ oooooooooo ]<[ooo/ooo\]>) \\\ 2x Promotion-
{ooo}/ooo //([ooooooooo/ooo\/ooo\/ooo\/o\]<[ooo/ooo\]>) \\\ -Carry- and 3x -Collection-
{ooo}/ooo//([ooo/ooo\]<[ooo/ooo\]>)(dito)(dito)([/o\]<[ooo/ooo\]>) \\\ 3x Distribution of <[ooo/ooo\]>
{ooo}/ooo//( )( )( )([/o\]<[ooo/ooo\]>) \\\ 3x Inversion of [ooo/ooo\]
{ooo}/ooo//ooo ([//oooooooooo\\]<[ooo/ooo\]>) \\\ -Carry-
{ooo}/ooo//ooo //([ oooooooooo ]<[ooo/ooo\]>)\\\\\ 2x Promotion-
...

Real part of the complex exponential function

Imaginary part of the complex exponential function
Transcendental
J = [<o>] Phase Element Definition ln -1 = πi (~3.1415926535i)
(J) = ([<o>]) = <o> Involution2 eJ = eπi = -1
(o) Euler number e1 = e (~2.71828182845)
((o)) ee (~15.1542622414)
[<(a)>] = a[<o>] Phase Independence of a Theorem 7 ln -ea = a + ln -1
(a JJ ) = (a) J-cancellation1 Theorem 8 ea + 2*ln -1 = ea + ln (-1)² = ea + ln 1 = ea = ea + 2πi
(a<bJJ>) = (a<b>) J-cancellation2 Theorem 9 ea - b - 2*ln -1 = ea - b - ln (-1)² = ea - b - ln 1 = ea - b = ea - b - 2πi
(J) = (<J>) Self-Inversion of J Theorem 10 eJ = e-J
(([J]<[oo]>)) i and <i> eJ/2 = √-1 = ±i
i = (<[<i>]>) = <(<[i]>)> imaginary unit Theorem 11 i = 1/-i = -e-ln i
([<J>][i]) = ([J][<i>]) Circle number Pi -J*i = J/i = J*-i = π (~3.1415926535)
Like ln in the imaginary, by an arbitrary multiple of 2π is ambiguous, so at e all values are repeated every 2πi.
To represent this, the J-cancellation and self-inversion of J is permissible only within an instantiation,
or they must be contained in at least one more instantiation than in abstractions.
Proof of Phase Independence of a for a != [∎]
[<(a)> ]
[<(a)>( [ ])] -Involution2
[<(a)>(a[ ])] -Absorption of a
[<(a)>(a[o <o>])] -Inversion of o
[<(a)>(a[o])(a[<o>])] Distribution of a
[<(a)>(a )(a[<o>])] Involution1
[ (a[<o>])] Inversion of (a)
a[<o>] Involution1
Proof of J-cancellation within ()
[<o>][<o>]
[<([<o>])>] -Phase Independence von [<o>]
[< <o> >] Involution2
[ o ] Inverse Cancellation
Involution1
Proof of J-cancellation1 within ()
(a[<o>][<o>]) -Involution1
<<(a[ o ][ o ])>> 2x Inverse Promotion1
(a[ o ][ o ]) Inverse Cancellation
(a ) 2x Involution1
Proof of Self-Inversion of J within ()
(J )
(J< >) -Inversion of void
(J<J J>) -J-cancellation2
(J<J><J>) -Inverse Collection
( <J>) Inversion of J
f1(x) = x2
f2(x) = √x
f1(f2(x)) = x (√x)2 = x
f2(f1(x)) = ±x √(x2) = ±x

Real part and imaginary part of the complex square root

Real part and imaginary part of the complex cube root
Proof of ambiguity of (([J]<[oo]>)) or √-1
Whenever the exponent is a non-integer rational number, there is ambiguity.
And if the exponent is less than or equal to 0, the base equals 0 gives a degenerate point.
i * i = -i * -i = (±i)2 = -1
([(([J]<[oo]>))][(([J]<[oo]>))])
(([[(([J]<[oo]>))]][oo])) Cardinality of [(([J]<[oo]>))]
(( [J]<[oo]> [oo])) 2x Involution1
(( [J] )) Inversion of [oo]
<o> 2x Involution2
and
([<(([J]<[oo]>))>][<(([J]<[oo]>))>])
<<([ (([J]<[oo]>)) ][ (([J]<[oo]>)) ])>> 2x Inverse Promotion1
([ (([J]<[oo]>)) ][ (([J]<[oo]>)) ]) Inverse Cancellation
(([[(([J]<[oo]>))]][oo]) ) Cardinality of [(([J]<[oo]>))]
(( [J]<[oo]> [oo]) ) 2x Involution1
(( [J] ) ) Inversion of [oo]
<o> 2x Involution2
Proof of imaginary unit1
(<[<<(([ J ]<[oo]>))>>]>)
(< ([ J ]<[oo]>) >) Inverse Cancellation, Involution1
( ([<J>]<[oo]>) ) -Inverse Promotion1
( ([ J ]<[oo]>) ) -Self-Inversion of J
Proof of imaginary unit2
<(<[(([ J ]<[oo]>))]>)>
<(< ([ J ]<[oo]>) >)> Involution1
<( ([<J>]<[oo]>) )> -Inverse Promotion1
<( ([ J ]<[oo]>) )> -Self-Inversion of J
-1 * -1 = 1
([<o>][<o>])
( ) J-cancellation1
ee = ee
(([[(o)]][(o)]))
(( o )) 3x Involution1
< a > = ([<([a])>]) = ([a][<o>]) = ([a]J) 2x-Involution2, Phase Independence of [a]
[< a >] = [<([a])>] = [a][<o>] = [a]J -Involution2, Phase Independence of [a]
<( a )> = ([<( a )>]) = (a[<o>]) = (aJ) -Involution2, Phase Independence von a
Indefiniteness of Cardinality of J within ()
... = (<JJJJ>) = (<JJ>) = o = (JJ) = (JJJJ) = ... J-cancellation1, J-cancellation2, Inversion of void
... = (<JJJ>) = (<J>) = (J) = (JJJ) = (JJJJJ) = ... J-cancellation1, J-cancellation2, Self-Inversion of J
Indefiniteness of [∎] within ()
( [∎] )
( [∎] JJ) -J-cancellation1
([<([∎])>]J) -Phase Independence of [∎]
([< ∎ >]J) Involution2 - Indefiniteness der Inversion of ∎
Indefiniteness of 00
(([∎]∎))
(( ∎)) Absorption of [∎]
( ) Involution2
=> 00 = 1 but from the Indefiniteness of [∎] follows in principle the Indefiniteness of 00
a0 = 1
(([[a]]∎))
(( ∎)) Absorption of [[a]]
( ) Involution2
0n = 0 n ∈ ℕ
(([∎][o...o]))
(∎...∎) Cardinality of ∎
(∎ ) Absorption of ∎
Involution2
ln 0 + ln -1 = ln 0
∎J
[<(∎)>] -Phase Independence of ∎
[< >] Involution2
[ ] Inversion of void
Proof of multiplicative Self-Inversion of (([J]<[oo]>)) (like o)
1/±i = ∓i
(<[(([ J ]<[oo]>))]>)
(< ([ J ]<[oo]>) >) Involution1
( ([<J>]<[oo]>) ) -Inverse Promotion1
( ([ J ]<[oo]>) ) -Self-Inversion of J
and
(<[<(([ J ]<[oo]>))>]>)
<(<[ (([ J ]<[oo]>)) ]>)> -Inverse Promotion2
<(< ([ J ]<[oo]>) >)> Involution1
<( ([<J>]<[oo]>) )> -Inverse Promotion1
<( ([ J ]<[oo]>) )> -Self-Inversion of J
∓i = ±i
<(([ J ]<[oo]>))>
(<[<<(([ J ]<[oo]>))>>]>) imaginary unit11
(< ([ J ]<[oo]>) >) Inverse Cancellation, Involution1
( ([<J>]<[oo]>) ) -Inverse Promotion1
( ([ J ]<[oo]>) ) -Self-Inversion of J
±i = ∓i
(([ J ]<[oo]>))
<(<[(([ J ]<[oo]>))]>)> imaginary unit12
<(< ([ J ]<[oo]>) >)> Involution1
<( ([<J>]<[oo]>) )> -Inverse Promotion1
<( ([ J ]<[oo]>) )> -Self-Inversion of J
J/i = i*π / i = π
([J]<[(([ J ]<[oo]>))]>)
([J]< ([ J ]<[oo]>) >) Involution1
([J] ([<J>]<[oo]>) ) -Inverse Promotion1
([J] ([ J ]<[oo]>) ) -Self-Inversion of J
and
([ J ]<[<(([ J ]<[oo]>))>]>)
<([ J ]<[ (([ J ]<[oo]>)) ]>)> Inverse Promotion3
([<J>]<[ (([ J ]<[oo]>)) ]>) -Inverse Promotion1
([<J>]< ([ J ]<[oo]>) >) Involution1
([<J>] ([<J>]<[oo]>) ) -Inverse Promotion1
([<J>] ([ J ]<[oo]>) ) -Self-Inversion of J
J * -i = π
([ J ][<(([ J ]<[oo]>))>])
<([ J ][ (([ J ]<[oo]>)) ])> Inverse Promotion1
([<J>][ (([ J ]<[oo]>)) ]) -Inverse Promotion1
([<J>] ([ J ]<[oo]>) ) Involution1
and
([J][<<(([ J ]<[oo]>))>>])
([J][ (([ J ]<[oo]>)) ]) Inverse Cancellation
([J] ([ J ]<[oo]>) ) Involution1
π * i = J
([([<J>][i])][i])
( [<J>][i] [i]) Involution1
( [<J>][<o> ]) i * i = -1
<<( [ J ][ o ])>> 2x Inverse Promotion1
J Inverse Cancellation, Involution1, 2x Involution2
ii = eJ*i/2 = e-π/2 = e-π/2-2πk k ∈ ℤ
(([[(([J]<[oo]>))]][(([J]<[oo]>))]))
(( [J]<[oo]> ([J]<[oo]>) )) 3x Involution1
and
( ([[(([ J ]<[oo]>))]][<(([J]<[oo]>))>]) )
(<( [ J ]<[oo]> [ (([J]<[oo]>)) ])>) 2x Involution1, Inverse Promotion1
( ( [<J>]<[oo]> [ (([J]<[oo]>)) ]) ) -Inverse Promotion1
( ( [ J ]<[oo]> ([J]<[oo]>) ) ) Involution1, -Self-Inversion of J
i-2i = eπ
(([[(([J]<[oo]>))]][(([J]<[oo]>))][oo]))
(( [J]<[oo]> ([J]<[oo]>) [oo])) 3x Involution1
(( [J] ([J]<[oo]>) )) Inversion of [oo]
and
( ([[(([ J ]<[oo]>))]][<(([J]<[oo]>))>][oo]) )
(<( [ J ]<[oo]> [ (([J]<[oo]>)) ][oo])>) 2x Involution1, Inverse Promotion1
( ( [<J>]<[oo]> [ (([J]<[oo]>)) ][oo]) ) -Inverse Promotion1
( ( [ J ] ([J]<[oo]>) ) ) Involution1, -Self-Inversion of J
1/√a = √(1/a)
(<[(([ [a] ]<[oo]>))]>)
(< ([ [a] ]<[oo]>) >) Involution1
( ([<[a]>]<[oo]>) ) -Inverse Promotion1
√9 = √±32 = ±3
(([[ ooooooooo ]]<[oo]>))
(([[( [ooo][ooo] )]]<[oo]>)) Cardinality of ooo
(([[(([[ooo]][oo]))]]<[oo]>)) Cardinality of [ooo]
(( [[ooo]][oo] <[oo]>)) 2x Involution1
(( [[ooo]] )) Inversion of [oo]
ooo 2x Involution2
and
(([[ ooooooooo ]]<[oo]>))
(([[ ( [ ooo ][ ooo ]) ]]<[oo]>)) Cardinality of ooo
(([[<<( [ ooo ][ ooo ])>>]]<[oo]>)) -Inverse Cancellation
(([[ ( [<ooo>][<ooo>]) ]]<[oo]>)) 2x -Inverse Promotion1
(([[ (([[<ooo>]][oo]) ) ]]<[oo]>)) Cardinality of [<ooo>]
(( [[<ooo>]][oo] <[oo]>)) 2x Involution1
(( [[<ooo>]] )) Inversion of [oo]
<ooo> 2x Involution2
-22 = 22
(([[<oo>]][oo]))
([<oo>][<oo>]) -Cardinality of [<oo>]
<<([ oo ][ oo ])>> 2x Inverse Promotion1
([ oo ][ oo ]) Inverse Cancellation
(([[oo]][oo])) Cardinality of [oo]
ab = -ab
(( [[ a ]] [b]))
(( [[ a ]][oo] <[oo]>[b])) -Inversion of [oo]
(([[ (([[ a ]][oo])) ] ]<[oo]>[b])) 2x -Involution1
(([[ ( [ a ][ a ] ) ] ]<[oo]>[b])) -Cardinality of [a]
(([[<<( [ a ][ a ] )>>] ]<[oo]>[b])) -Inverse Cancellation
(([[ ( [<a>][<a>] ) ] ]<[oo]>[b])) 2x -Inverse Promotion
(([[ (([[<a>]][oo])) ] ]<[oo]>[b])) Cardinality of [<a>]
(( [[<a>]][oo] <[oo]>[b])) 2x Involution1 !!! ambiguity
(( [[<a>]] [b])) Inversion of [oo]
Self-Inversion problem
-0 = 0
1/1 = 1
<∎>
∎ = ∎∎ = ∎∎∎ = ∎∎∎∎ = ...
J = <J> ... = <JJJJ> = <JJ> = = JJ = JJJJ = ... , ambiguity of (([J]<[oo]>))
1/±i = ∓i
ab = -ab
Three-dimensional representation of the Eulerian formula ez*i = cos z + i sin z => (([z]([J]<[oo]>)))

Real part of the complex sine function
Imaginary part of the complex sine function
an+1 = ([an]J) o -> (J) -> o -> (J) -> o -> ... <= (1 -> -1 -> 1 -> -1 -> 1 -> ...)
an+1 = ([an][i]) o -> i -> (J) -> <i> -> o -> ... <= (1 -> i -> -1 -> -i -> 1 -> ...)
an+1 = <([an][i])> o -> <i> -> (J) -> i -> o -> ... <= (1 -> -i -> -1 -> i -> 1 -> ...)
ea*i = cos a + i sin a => (([a][i]))
e-a*i = cos a - i sin a => (([<a>][i]))
sin a = (ea*i - e-a*i)/2i => ([ (([a][i]))<(([<a>][i]))>]<[oo][i]>)
cos a = (ea*i + e-a*i)/2 => ([ (([a][i])) (([<a>][i])) ]<[oo] >)
ea = cosh a + sinh a => (a)
e-a = cosh a - sinh a => (<a>)
sinh a = (ea - e-a)/2 => ([(a)<(<a>)>]<[oo]>)
cosh a = (ea + e-a)/2 => ([(a) (<a>) ]<[oo]>)
sin(x + i y) = sin(x) cosh(y) + i cos(x) sinh(y)
cos(x + i y) = cos(x) cosh(y) - i sin(x) sinh(y)
sin(z) = -i sinh(i z)
sinh(z) = -i sin(i z)
cos(z) = cosh(i z)
cosh(z) = cos(i z)
sin'(z) = cos(z)
sinh'(z) = cosh(z)
cos'(z) = -sin(z)
cosh'(z) = sinh(z)
unit circle x2 + y2 = 1 (([[x]][oo])) (([[y]][oo])) = o
unit hyperbole x2 - y2 = 1 (([[x]][oo]))<(([[y]][oo]))> = o